Kunkel, William2019-02-122019-02-122018-10https://hdl.handle.net/11299/201689University of Minnesota Ph.D. dissertation.October 2018. Major: Electrical Engineering. Advisor: James Leger. 1 computer file (PDF); xv, 153 pages.This thesis addresses two topics in coherent beam combining. First, a fiber laser with two parallel gain elements is considered, where the phase difference between beams is passively adjusted by the difference in gain, via Kramers-Kronig phase. This addresses an important aspect of beam combining, whereby phase agreement is attained between constituent beams. The second part of this thesis is devoted to beam shaping with custom gradient-index optics. Control over the spatial characteristics of laser beams is essential for efficient combining, as overlapped beams must interfere constructively across their entire transverse profile. An experiment is conducted to isolate and measure the Kramers-Kronig phase of the fiber laser mode. Measurements support a nonlinear model of passive phasing in the resonator, showing that the gain self-adjusts in response to externally applied phase errors between the arms of the resonator. The effect can be harnessed to achieve efficient beam combining without external feedback to the laser array. We also describe a numerical design method to create spatially-varying refractive index functions that accomplish arbitrary beam transformations in two and three dimensions. The refractive index designs compensate the effects of diffraction, overcoming the limitations of traditional optics such as aspheric lenses and diffractive optics. Example designs, such as a Gaussian-to-supergaussian mode converter and an array beam combiner, are shown. A device fabricated by femtosecond laser writing is tested experimentally.enbeam shapingfiber lasersGRINlaser arrayslaser beam combiningphase retrievalThesis or Dissertation